Random algebraic construction of extremal graphs

Bukh, Boris

doi:10.1112/blms/bdv062

Cited by 46 publications

(84 citation statements)

References 23 publications

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“…That is, the events are independent. This observation is again due to Bukh and corresponds to Lemma of his paper , though we state and prove it in greater generality. Lemma Suppose

q > (\frac{0pt}{m})

and

d ⩾ m - 1

.…”

Section: Preliminariesmentioning

confidence: 54%

“…We will now determine the probability that a randomly chosen polynomial from P d passes through a given set of points. For one point, the probability is 1/q, as shown by the following simple result of Bukh [6]. We include the proof for completeness.…”

Section: Preliminariesmentioning

confidence: 82%

“…The final ingredient we require is again essentially due to Bukh and says that if

W

is a variety which is defined over

F_{q}

, then there is a finite collection of absolutely irreducible varieties

Y_{1}, \dots, Y_{s}

, each of which is defined over

F_{q}

, such that

\cup_{i = 1}^{s} Y_{i} false(F_{q} false) = W false(F_{q} false)

. Since it is less standard than the previous two lemmas, we include the proof.…”

Section: Preliminariesmentioning

confidence: 99%

“…If

T ({double-struckF}_{q})

were equal to

S

, we could apply Lemma from to show that

S

is either bounded by a constant or quite large and then use the corollary of Markov's inequality proved above to show that there are very few pairs

(w_{1}, w_{2})

for which

S

is large. Unfortunately,

T ({double-struckF}_{q})

may contain degenerate walks as well as the paths we are interested in, so we must somehow take these into account.…”

Section: The Constructionmentioning

confidence: 99%

“…Very recently, Bukh found a simple, elegant method for showing that the Kővári–Sós–Turán bound is tight for

t

sufficiently large in terms of

s

, fusing the algebraic techniques used in all previous constructions with an application of the probabilistic method. In this paper, we adapt his method to make progress on an equally stubborn problem.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Graphs with few paths of prescribed length between any two vertices

Conlon

2019

Bull. London Math. Soc.

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We use a variant of Bukh's random algebraic method to show that for every natural number k⩾2 there exists a natural number ℓ such that, for every n, there is a graph with n vertices and normalΩkfalse(n1+1/kfalse) edges with at most ℓ paths of length k between any two vertices. A result of Faudree and Simonovits shows that the bound on the number of edges is tight up to the implied constant.

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“…That is, the events are independent. This observation is again due to Bukh and corresponds to Lemma of his paper , though we state and prove it in greater generality. Lemma Suppose